Gradient function of matlab

Can someone please explain how gradient function works? (say, how is dx1(1,2)==0.5)

>> a=[1 3 2 11 18; 7 14 9 5 10; 15 7 13 18 9; 19 12 17 7 14 ]

a =

     1     3     2    11    18
     7    14     9     5    10
    15     7    13    18     9
    19    12    17     7    14

>> [dx1 dy1]=gradient(a)

dx1 =

    2.0000    0.5000    4.0000    8.0000    7.0000
    7.0000    1.0000   -4.5000    0.5000    5.0000
   -8.0000   -1.0000    5.5000   -2.0000   -9.0000
   -7.0000   -1.0000   -2.5000   -1.5000    7.0000

dy1 =

    6.0000   11.0000    7.0000   -6.0000   -8.0000
    7.0000    2.0000    5.5000    3.5000   -4.5000
    6.0000   -1.0000    4.0000    1.0000    2.0000
    4.0000    5.0000    4.0000  -11.0000    5.0000

Answer 1

[David Young]

The basic operation is to take half the difference between the two values on either side of the point you are considering. For example,

 dx1(2,2) = 0.5 * (a(2,3) - a(2,1))   (i.e. 0.5*(9-7))

and

 dx2(2,2) = 0.5 * (a(3,2) - a(1,2))  (i.e. 0.5*(7-3))

So for your example, dx1(1,2) = 0.5 * (2 - 1).

This has to be a little different for points on the edge of the matrix when a neighbouring value is not available. In these cases the single sided difference is taken, so for example

 dx1(3,1) = a(3,2) - a(3,1)  (i.e. 7-15)

If you are familiar with convolution, dx1 is just

 conv2(a, [0.5 0 -0.5])

except for the left and right columns, and dy1 is

 conv2(a, [0.5 0 -0.5].')

except for the top and bottom rows.